Tikhonov-Phillips Regularization of the Radon Transform

  • Alfred K. Louis
Part of the International Series of Numerical Mathematics book series (ISNM, volume 73)

Abstract

With the help of the singular value decomposition of the Radon transform the Tikhonov-Phillips regularization with suitable Sobolev norm is analytically computed. This allows for studying the influence of the regularization norm. Even more important is the effect of the regularization parameter which can now be studied explicitly, and optimally selected dependent on data noise and additional information on the searched-for function.

Keywords

Regularization Parameter Null Space Data Error Radon Transform Minimum Norm Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Abramowitz, M., Stegem, I-A. (eds.) (1965) Handbook of mathematical functions ( Dover, New York ).Google Scholar
  2. [2]
    Cormack, A.M. (1964) Representation of functions by its line integrals, with some radiological applications II. J. Appl. Phys. 35, 2908–2913.CrossRefGoogle Scholar
  3. [3]
    Gradstheyn, I.S., Ryzhik, I.M. (1980) Table of integrals, series and products, 3rd edn. ( Academic Press, New York).Google Scholar
  4. [4]
    Hamming, R.W. (1977) Digital Filters (Prentice Hall, Englewood Cliffs, N.J.).Google Scholar
  5. [5]
    Herman, G.T. (1980) Image reconstruction from projections: the fundamentals of computerized tomography (Academic Press New York).Google Scholar
  6. [6]
    Jerry, A.J. (1977) The Shannon sampling theorem - its various extensions and applications: a tutorial review. Proc. IEEE 65, 1565–1596.CrossRefGoogle Scholar
  7. [7]
    Logan, B.F. (1975) The uncertainty principle in reconstruct ing functions from projections. Duke Math. J. 42, 661–706.CrossRefGoogle Scholar
  8. [8]
    Louis, A.K. (1984) Orthogonal function series expansions and the null space of the Radon transform. SIAM J. Math. Anal. 15, 621–633.CrossRefGoogle Scholar
  9. [9]
    Louis, A.K. (1984) Nonuniqueness in inverse Radon problem: the frequency distribution of the ghosts. Math. Z. 185, 429–440.CrossRefGoogle Scholar
  10. [10]
    Louis, A.K., Natterer, F. (1983) Mathematical problems of computerized tomography. Proc. IEEE 71, 379–389.Google Scholar
  11. [11]
    Ludwig, D. (1966) The Radon transform on Euclidean spaces. Comm. Pure Appl. Math. 19, 49–81.CrossRefGoogle Scholar
  12. [12]
    Marr, R.B. (1974) On the reconstruction of a function on a circular domain from sampling of its line integrals. J. Math. Anal. Appl. 19, 357–374.CrossRefGoogle Scholar
  13. [13]
    Natterer, F. (1980) A Sobolev space analysis of picture reconstruction. SIAM J. Appl. Math. 39, 402–411.CrossRefGoogle Scholar
  14. [14]
    Natterer, F. (1983) On the order of regularization methods. In: Hammerlin, G., Hoffmann, K.-H. (eds.) Improperly posed problems and their numerical treatment. ( Birkhauser, Basel, Boston, Stuttgart ), 189–203.Google Scholar
  15. [15]
    Radon, J. (1917) Uber die Bestimmung von Funktionen durch ihre Integralwerte langs gewisser Mannigfaltigkeiten. Ber. Verh. Sachs. Akad. Wiss. Leipzig 69, 262–277.Google Scholar
  16. [16]
    Smith, K.T., Solmon, D.C., Wagner, S.L. (1977) Practical and mathematical aspects of reconstructing objects from radiographs. Bui. AMS 83, 1227–1270.CrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel 1985

Authors and Affiliations

  • Alfred K. Louis
    • 1
  1. 1.Fachbereich MathematikUniversität KaiserslauternKaiserslauternGermany

Personalised recommendations